Three-way group decisions with interval-valued decision-theoretic rough sets based on aggregating inclusion measures

Abstract

The main purpose of three-way decisions with interval-valued decision-theoretic rough sets is to make decisions by minimizing interval-valued expected loss functions. Since the set of interval-valued expected losses is not a totally ordered set, a new mechanism is presented to make three-way group decisions with interval-valued decision-theoretic rough sets by calculating inclusion measures between two arbitrary interval-valued expected loss functions. Firstly, based on conjunctive and disjunctive semantics of intervals, inclusion measures are proposed based on the partial orders via different semantics, respectively. Secondly, the framework of three-way decisions with interval-valued decision-theoretic rough sets is presented based on inclusion measures of intervals. Thirdly, three-way decisions with interval-valued decision-theoretic rough sets are extended to group decisions. To improve the precision of interval-valued loss evaluation for the final group decisions, we divide interval-valued expected loss functions for each expert into several trivial intervals, then the matrix of inclusion measures for interval-valued expected loss functions between every two experts is obtained by aggregating all the inclusion measures. Fourthly, we employ the methodology of three-way decisions with interval-valued decision-theoretic rough sets and select rules with minimal costs or risks to obtain the optimal decision rules of group decisions. To highlight the performance of our methodology, by using score functions to transform an interval into a real one we introduce another method based on the principle of justifiable granularity to obtain decision rules of group decisions. Finally several data sets are employed to evaluate our methodology.